The sparsity of a KR rearrangement \(S\) 
that pushes forward a density \(\pi \)  on
\(\mathbb {R}^n\)  to \(\mathcal {N}({\bf 0}, {\bf I}_n)\)  depends
on marginal conditional independence properties of
\(\pi \) .
As explained in [1]}, if \(Z=(Z_1,\ldots ,Z_n)\sim \pi \) , then the \(k\) th component of \(S\)  is not a
function of the \(j\) th input variable (\(j<k\) ) if
\(Z_j\)  and \(Z_k\)  are conditionally independent given
the remaining variables with index less than \(k\) , i.e.,
if
\( Z_j  \perp \!\!\!\perp Z_k \vert Z_{ \lbrace 1:k-1\rbrace \setminus \,j}.\) 
